Question

advanced Calc

need help with 3rd bullet point

In the following problems, justify all answers rigorously by appropriate application of the intermediate value theorem and mean value theorem and whatever else you need • Prove that the polynomial x^3 - 3x - 1 has exactly three roots. • Prove that the polynomial x^3 - 3x + 4 has exactly one root . Find the value of such that the polynomial x^3 - 3x + c has exactly two roots

Answer 1

Given polynomial P(x) = x²-3x+6; P' (m) = 3 x 3 = 3 (421) = 3(m-1)(x+1). sign scheme + un (-00,-1) polynomial ( curre) increasing (-1, 1) curore of polynomial decreasing (1,0) curve of polynomicy inereasing, P10)=c

C=plo) (0,0). + + ti 0 1 ? Praco nati (critical points) By Lagrange mean ralue theorem; if cyo then the curine intersect at two points on and may on the 2. axis. Therefore, if cyo then the pe polynomial peny has exactly two rreas Roots in the Rarge (-1,0).